Research Article Microstructure, Strength, and Fracture … · 2017. 7. 24. · InternationalJournalofMetals 1.35 1.30 1.25 1.20 1.15 1.10 r2 = 0.99 Average grain size Fracture surface

Research ArticleMicrostructure, Strength, and Fracture TopographyRelations in AISI 316L Stainless Steel, as Seen through a FractalApproach and the Hall-Petch Law

Oswaldo Antonio Hilders,1 Naddord Zambrano,2 and Ramón Caballero3

1Department of Physical Metallurgy, School of Metallurgical Engineering and Materials Science, Central University of Venezuela,Apartado 47514, Los Chaguaramos, Caracas 1041-A, Distrito Capital, Venezuela2Foundation for Professional Development, The Venezuelan College of Engineering, Caracas 1050, Venezuela3Failure Analysis Laboratory, School of Metallurgical Engineering and Materials Science, Central University of Venezuela,Apartado 47514, Los Chaguaramos, Caracas 1041-A, Distrito Capital, Venezuela

Correspondence should be addressed to Oswaldo Antonio Hilders; [email protected]

Received 8 May 2015; Accepted 1 July 2015

Academic Editor: Carlos Garcia-Mateo

Copyright © 2015 Oswaldo Antonio Hilders et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The influence of the fracture surface fractal dimension𝐷𝐹and the fractal dimension of grain microstructure𝐷

𝑀on the strength of

AISI 316L type austenitic stainless steel through the Hall-Petch relation has been studied. The change in complexity experimentedby the net of grains, as measured by 𝐷

𝑀, is translated into the respective fracture surface irregularity through 𝐷

𝐹, in such a way

that the higher the grain size (lower𝐷𝑀values) the lower the fracture surface roughness (lower values of𝐷

𝐹) and the shallower the

dimples on the fractured surfaces. The material was heat-treated at 904, 1010, 1095, and 1194∘C, in order to develop equiaxed grainmicrostructures and then fractured by tension at room temperature. The fracture surfaces were analyzed with a scanning electronmicroscope, 𝐷

𝐹was determined using the slit-island method, and the values of 𝐷

𝑀were taken from the literature. The relation

between grain size, 𝐷𝑀, mechanical properties, and 𝐷

𝐹, developed for AISI 316L steel, could be generalized and therefore applied

to most of the common micrograined metal alloys currently used in many key engineering areas.

1. Introduction

Many steels and conventional metallic alloys in general stillfill an important place in engineering technology. Althoughnanocrystalline materials show promise for applications inseveral fields [1–4], their use is generally restricted for large-scale applications [3]. On the other hand, many importantengineering applications of materials involve the use ofconventionalmetallic alloys in polycrystalline form. For thesealloys, the individual grains generally ranged between 10and 300 𝜇m. Conventional metallic alloys are widely usedin several engineering areas in which they will be difficultto replace in the near future. The knowledge related withmicroscopic grained metallic alloys is constantly updated.Some examples of these alloys can be seen in the recentliterature [5–13].

Inmetallic polycrystalline alloys, the relation between thefracture surface features and the underlying microstructureis very well known [14–16]. As the mechanical propertiesdepend on the microstructure, it is clear that the topographyof the fractured surfaces is also related to the mechanicalproperties. On the other hand, in view of the usefulness ofthe fractal geometry to study the relation between fracturesurface tortuosity andmechanical properties [17–27] and thatof the Hall-Petch relationship to relate microstructure andmechanical properties [28–32], it is understandable that themicrostructure-fracture topography-mechanical propertiesrelationship can be studied by combining both approaches.So far only a few bridges have been built between thesetwo approaches (see, e.g., [28, 33]). The aim of this workis to establish a quantitative correlation between the frac-ture surface fractal dimension 𝐷

𝐹(a measure of tortuosity

Hindawi Publishing CorporationInternational Journal of MetalsVolume 2015, Article ID 624653, 10 pageshttp://dx.doi.org/10.1155/2015/624653

2 International Journal of Metals

of a fracture surface) and the fractal dimension of grainmicrostructure 𝐷

𝑀(a measure of complexity of the internal

net of grains) in AISI 316L type steel, as both can be relatedthrough theHall-Petch law.The link between𝐷

𝑀and𝐷

𝐹can

provide an understanding of the role of microstructure onthe mechanics of crack propagation. This link arises becausemicrostructure has a major influence on the topography ofthe fracture surface. The correlation between 𝐷

𝐹and 𝐷

𝑀

could be a useful tool to analyze the connection amongmicrostructure, design, fabrication, and performance, in bothconventional [28–32] and nanocrystallinemetallic alloys [1, 2,4, 34, 35].

2. Materials and Methods

The material used in this work was austenitic type AISI 316Lstainless steel fabricated into hot-rolled bar with diameter of25.4mm provided by a commercial supplier. The chemicalcomposition of the steel is 16.9Cr, 12.0Ni, 2.52Mo, 1.5Mn,0.35Si, 0.025C, 0.035N, 0.030P, and 0.030S (wt.%). Four slicesand eight tensile samples of 25.4mm gage length weretaken from the as-received bar and heat-treated at fourdifferent temperatures: 904, 1010, 1095, and 1194∘C, in orderto develop equiaxed grain microstructures (one slice andtwo tensile samples for each temperature). The temperatureswere selected according to the work of Colás [36]. After theheat treatments, the slices and tensile samples were water-quenched at room temperature. Then, the slices were groundand polished by standard metallographic methods, while themicrostructure was revealed by electrolytic etching.

An automatic image analyzer was used in order toperform grain size measurements according to the meanlinear intercept method. At least ten different fields of viewwere analyzed for each metallographic sample. Before per-forming themeasurements of grain size, the grain boundarieswere extracted and enhanced by means of image processingtechniques [37]. Briefly, well-defined grain boundaries wereobtained transforming our 256 gray level images to two grayvalues: black and white (thresholding). Then, a specializedoperation that prevents the separation of grain boundarieswhile eroding away pixels is performed (skeletonization).Finally, an image processingwas done to eliminate impurities,particles, and so forth in the grain interiors (hole filling).

The tensile samples were deformed at room temperatureat a nominal strain rate of 3.5 × 10−4/s in an Instron tensilemachine until fracture. A 10mm section from both the cupand the cone portions of fractured tension samples wasremoved and the fracture surfaces were analyzed with ascanning electron microscope (SEM), which was operated at20Kv. The fractographic features were studied in the centralregion of the cup portion of broken samples using severalmicrographs for each case.

The values of the fracture surface fractal dimension 𝐷𝐹

have been determined using the so-called slit-island method(SIM) [17, 19, 24, 38–41] in the central region of the respectivecone portions. Each conewas coldmolding using epoxy resin,which was pouring over the sample (which was previouslyattached to a cylindrical support of convenient size). Each

Table 1: Average grain size and𝐷𝐹data.

𝑇 (∘C) 904 1010 1095 1194𝑑 (𝜇m) 21.4 ± 3.1 27.2 ± 3.3 67.9 ± 4.9 148.8 ± 7.8𝐷𝐹

1.281 1.260 1.225 1.142

sample was positioned face up, allowing the epoxy to cover allthe fracture surface. Grinding and polishing operations wereperformed parallel to the mean plane of fracture, developinga number of successive layers in which part of the fracturesurface becomes visible (“islands”). As the layers increasedin number, the islands do, and growth and coalescence ofislands take place. For a particular 𝑗th layer with 𝑛 islands,𝑃𝑖 and 𝐴𝑖 represent the perimeter and the area of the 𝑖thisland, respectively. Taking into account all the islands in thislayer, the total perimeter and the total area are Σ𝑃𝑖 and Σ𝐴𝑖,respectively. For all the layers, a full logarithmic scale diagramof Σ𝑃𝑖 versus Σ𝐴𝑖 leads to obtaining a straight curve, fromwhich 𝐷

𝐹= 2 × slope. Figure 1 shows an example of a

sequence of 4 nonconsecutive partial layers (out of 26), tocalculate the value of𝐷

𝐹= 1.142.

On the other hand, the values of the microstructuralfractal dimension were taken from the work of Colás [36].In order to estimate the values of 𝐷

𝑀, Colás employed the

box-counting method [42]. In this method, a square gridcontaining boxes of a given side length ℎ is superimposed onthe grain boundary pattern.Then, the number of boxes𝑁(ℎ)containing boundary contours is counted. This process isrepeated to find𝑁(ℎ) for smaller values of ℎ. Asymptotically,in the limit of small ℎ,

𝑁(ℎ) = 𝑁𝑜ℎ−𝐷𝑀 , (1)

where 𝑁𝑜is a constant. For a fractal pattern, the slope of

the straight curve log𝑁(ℎ) versus log ℎ is the microstructuralfractal dimension𝐷

𝑀whose values are between 1.0 and 2.0.

3. Results and Discussion

3.1. The Relation between Yield Stress and 𝐷𝐹

3.1.1. Grain Size-𝐷𝐹

Relationship. Figure 2 shows opticalmicrographs of the microstructures of AISI 316L steel heat-treated at four different temperatures. Figure 3 shows theenhanced microstructures of AISI 316L steel after the imageprocessing.

The data of fracture surface fractal dimension 𝐷𝐹and

average grain size 𝑑 are listed in Table 1. On the other hand,Figure 4 shows the fracture surface fractal dimension 𝐷

𝐹

plotted against the average grain size 𝑑. As can be seen, thereis a negative linear correlation between 𝐷

𝐹and 𝑑, that is,

higher fracture surface fractal dimension for lower averagegrain size. The corresponding equation is

𝐷𝐹= 𝜆−𝛽𝑑, (2)

where 𝜆 = 1.30 and 𝛽 = 0.001/𝜇m. Equation (2) representsthe connection between the microstructure (grain size) andthe irregularity of the fracture surface (measured by𝐷

𝐹).The

International Journal of Metals 3

(a)

300𝜇m

(b)

(c) (d)

Figure 1: Optical micrographs of metallic islands of AISI 316L stainless steel, developed according to the SIM. (a) Layer number 1, (b) layernumber 7, (c) layer number 14, and (d) layer number 24.

results predicted in Figure 4 are consistent with the generalobservation that grain size reduction is a means to increasethe toughness of a metallic alloy [43, 44], since, as manystudies have been strongly supported, the higher the fracturesurface fractal dimension is, the higher the toughness is [19,21, 39, 45–47].

3.1.2. Fracture Surface Characteristics. Figure 5 shows sev-eral fractographs of the fractured tensile samples, whichcorrespond to the four heat treatment temperatures. Themicrovoid coalescence mechanism of separation was ob-served for all experimental conditions. Although the values of𝐷𝐹increase as the grain size decreases according to Figure 4,

the corresponding values of dimple size (as seen on the meanplane of fracture in Figure 5) were somewhat the same, whichimplies, in principle, that the dimple size (as measured by thesurface dimple diameter) is not related with the grain size. Inview of this fact, some factormust exist for the decrease in𝐷

𝐹

as the grain size increases.As 𝐷𝐹is a measure of the irregularity of the fracture

surface, it is suggested that the dimples become shallow as thegrain size increases (lower values of𝐷

𝐹) whichwas confirmed

by in situ extensive analysis (SEM). This can be checkedin Figure 5, at least for the extreme values of grain sizedeveloped in the present work: Figure 5(a): lower grain size(“deep dimples”) and Figure 5(d): higher grain size (“shallowdimples”).

The last view is supported by the fact that for smaller grainsize the plastic deformation spreads out to themicrostructuremore easily than for larger grain size (smaller grain materialstores more energy than larger grain material), creating arougher fracture surface (“deep dimples”) with a higherfractal dimension 𝐷

𝐹. The rougher the fracture surface,

the higher the stored energy and the tougher the material.Obviously, for this case the internal area of the “deep dimples”is also higher. Note that for a totally brittle material (which isnot the case in any of the studied conditions) the absorbedenergy is zero, and the fracture surface is flat.

Currently, a relationship between grain size and the deepof dimples in ductile fracture can be established indirectly,through the toughness. Note that, for a tougher materialtested in tension, the plasticity is higher and the dimplesare more enlarged in the axial direction (“deep dimples”).Figure 6 can illustrate these concepts.

On the other hand, the reason for dimples to remainabout the same diameter is that dimple size is controlledby the size and population of particles (precipitates and/orinclusions) in the interior of grains [48, 49]. After the nucle-ation of dimples begins from particles, their size increasesuntil the coalescence with other dimples, which inhibits anadditional growth. Provided the density and size of particleswere the same for all experimental conditions, the corre-sponding average dimple size becomes roughly the same.


100𝜇m

(a)

100𝜇m

(b)

100𝜇m

(c)

100𝜇m

(d)

Figure 2: Optical micrographs of microstructures of AISI 316L austenitic stainless steel heat-treated at (a) 904∘C, (b) 1010∘C, (c) 1095∘C, and(d) 1194∘C.

100𝜇m

(a)

100𝜇m

(b)

100𝜇m

(c)

100𝜇m

(d)

Figure 3:The processed images of grain microstructures of AISI 316L austenitic stainless steel corresponding to the micrographs of Figure 2.


1.35

1.30

1.25

1.20

1.15

1.10

r2 = 0.99

Average grain size

Frac

ture

surfa

ce fr

acta

l dim

ensio

nD

F

0 50 100 150

d (𝜇m)

Figure 4: Variation of the fracture surface fractal dimension withthe average grain size for AISI 316L austenitic stainless steel.

The relative absence of particles inside the dimples in Figure 5could be related with one or more of several factors: someparticles remain attached to the matting fracture surface,some particles were lost during the fracture event, or simplysome voids (few of them) nucleate homogeneously. Note thatthe relation between grain size and𝐷

𝐹is easier to explain for

the case of intergranular fracture (which was not obtained inany case in the present work). For intergranular separation, asthe path of the fracture surface follows the contour of grains,a lower grain size material will have a higher area of grainsand correspondingly a higher area of the fracture surface. Inthis case, the value of𝐷

𝐹will be higher too.

3.1.3. Hall-Petch Type Relation for 𝐷𝐹. The relation between

the fracture surface fractal dimension and mechanical prop-erties has been established through the Hall-Petch equation:

𝜎𝑦= 𝜎𝑜+ 𝑘𝑑−1/2, (3)

where 𝜎𝑦is the yield stress, 𝜎

𝑜is the friction stress which

opposes dislocation motion, 𝑘 is a constant related with thedifficulty in spreading yielding from grain to grain, and 𝑑is the average grain size. From (2) the average grain size is𝑑 = (𝜆−𝐷

𝐹)/𝛽, and then (3) is therefore rearranged to predict

the yield stress as

𝜎𝑦= 𝜎𝑜+ 𝑘(𝜆 −𝐷

𝐹)−1/2, (4)

where 𝑘 = 𝑘𝛽1/2 is a constant. For AISI 316L 𝜎𝑜= 163MPa

and 𝑘 = 0.77MPam1/2 [50], 𝑘 becomes 24.35MPa. Then,smaller grain size corresponds to higher fracture surfacefractal dimension and so to higher yield stress. Based on(4), two Hall-Petch type relations have been represented inFigure 7: a linear relation of 𝜎

𝑦versus (𝜆−𝐷

𝐹)−1/2, (𝜆 = 1.30),

and 𝜎𝑦versus 𝐷

𝐹. In the first case, the theoretical values

corresponding to (𝜆 − 𝐷𝐹)−1/2 ranged between 1.83 (𝐷

𝐹= 1)

and∞ (𝐷𝐹= 1.30), being the values of 𝜎

𝑦, 207.46MPa, and

∞, respectively.

Table 2: Average grain size and𝐷𝑀data [36].

𝑇 (∘C) 904 1010 1095 1194𝑑 (𝜇m) 19.6 ± 3.6 29.6 ± 4.8 59.3 ± 5.2 158.4 ± 8.2𝐷𝑀

1.504 1.470 1.423 1.365

The value of 𝜎𝑦= 207.46MPa represents the yield stress

for AISI 316L steel broken in tension, whose fracture surfaceis totally flat (𝐷

𝐹= 1). Theoretically, for this case the value

of 𝑑 should be 300𝜇m, (see (2)). For the curve 𝜎𝑦versus 𝐷

𝐹

in Figure 7, the limit conditions are the same, although theapproach to 𝜎

𝑦= ∞ as 𝐷

𝐹→ 1.30 is of a nonlinear nature.

The general Hall-Petch type relationship between 𝜎𝑦and 𝐷

𝐹

(see (4)) can be potentially useful to relate the fracture surfacefractal dimension 𝐷

𝐹with mechanical properties in many

commercial alloys.

3.2. The Relation between Yield Stress and 𝐷𝑀

3.2.1. Grain Size-𝐷𝑀

Relationship. According to Colás [36],the relationship between the average grain size 𝑑 and itsmicrostructural fractal dimension 𝐷

𝑀for AISI 316L can be

described by means of the following equation:

𝐷𝑀= 𝛼𝑑−𝜂, (5)

where 𝛼 = 1.716 𝜇m𝜂 and 𝜂 = 0.045. This equation predictsan increase in 𝐷

𝑀as the grain size decreases. The data

of microstructural fractal dimension and the range of theinvestigated average grain size upon which (5) was developedare listed in Table 2 [36].

3.2.2. Hall-Petch Type Relation for 𝐷𝑀. According to (5), the

average grain size is 𝑑 = (𝛼/𝐷𝑀)1/𝜂. This microstructural

fractal dimension dependence for 𝑑 is substituted into (3), topredict the yield stress as a function of𝐷

𝑀according to

𝜎𝑦= 𝜎𝑜+ 𝑘𝐷𝑀

1/(2𝜂), (6)

where 𝑘 = 𝑘/𝛼1/(2𝜂) is a new constant. From the values of𝑘, 𝛼, and 𝜂, 𝑘 = 1.91MPa. It can be seen that the yield stressincreases as the microstructural fractal dimension increases,which in turn means a decrease in the average grain size.For the present case, and based on (6), once again twoHall-Petch type relations can be plotted (Figure 8): a linearrelation 𝜎

𝑦versus (𝐷

𝑀)1/(2𝜂), (𝜂 = 0.045), and 𝜎

𝑦versus𝐷

𝑀.

Three subscales for the variable (𝐷𝑀)1/(2𝜂) have been used in

Figure 8 in order to preserve a natural arithmetic scale, whichfacilitates a good visualization of the fractal dimension values.This is performed according to the “level” of 𝐷

𝑀. The first

zone is defined for 1.0 ≤ (𝐷𝑀)1/(2𝜂)≤ 10, (1.0 ≤ 𝐷

𝑀≤ 1.23),

so from (6) two values of 𝜎𝑦can be defined, 𝜎

𝑦= 164.91MPa,

((𝐷𝑀)1/(2𝜂)= 1, 𝐷

𝑀= 1) and 𝜎

𝑦= 182.1MPa, ((𝐷

𝑀)1/(2𝜂)=

10, 𝐷𝑀= 1.23); thus, the curve for this zone can be traced.

It is suggested that the first zone could be identified with lowcomplex microstructures. The second zone ranged between(𝐷𝑀)1/(2𝜂)= 10 and some value around 100.The last limit can


10𝜇m

(a)

10𝜇m

(b)

10𝜇m

(c)

10𝜇m

(d)

Figure 5: Scanning electron fractographs of fractured tensile samples heat-treated at four different temperatures: (a) 904∘C: 𝑑 = 21.4 𝜇m,𝐷𝐹= 1.281; (b) 1010∘C: 𝑑 = 27.2 𝜇m,𝐷

𝐹= 1.260; (c) 1095∘C: 𝑑 = 67.9 𝜇m,𝐷

𝐹= 1.225; (d) 1194∘C: 𝑑 = 148.8 𝜇m,𝐷

𝐹= 1.142.

W

(a)

W

(b)

Figure 6: Ductile tension fracture surface profiles, showing the same average dimple size𝑊 (axial direction is ↕). (a) “Deep dimples” forsmaller grain size, in a tougher material with a higher𝐷

𝐹. (b) “Shallow dimples” for a higher grain size, in a material with smaller toughness

and lower𝐷𝐹.

movemore or less freely in a narrow range, since it representsan uncertainty of the value for (𝐷

𝑀)1/(2𝜂) (and therefore, the

value of 𝐷𝑀) from which the related microstructure starts

to be very complex in the third zone. For (𝐷𝑀)1/(2𝜂)= 100,

𝐷𝑀= 1.51, so for the second zone, 1.23 ≤ 𝐷

𝑀≤ 1.51. The

second zone can represent microstructures with an averagecomplexity. The curve for this zone can be traced using (6)for any two values of the corresponding scale. The curve forthe third zone can be defined using onemore time (6) and anytwo values of the third scale, for example, (𝐷

𝑀)1/(2𝜂)= 2, 212

(𝐷𝑀= 2), which corresponds to 𝜎

𝑦= 4, 387.84MPa, and

(𝐷𝑀)1/(2𝜂)= 2, 000 (𝐷

𝑀= 1.98), for 𝜎

𝑦= 3, 983MPa.

The value of 𝜎𝑦= 164.91MPa represents the yield stress

for AISI 316L steel broken in tension, for a grain size of≈162,740.33 𝜇m ≈ 16 cm (see (5)). We can write, as a firstapproximation (see (6)), that for 𝐷

𝑀= 1, 𝜎

𝑦≈ 𝜎𝑜=

163MPa. Correspondingly, we could consider thematerial asan individual grain of 16 cm (an infinite system as comparedto our real grains), which is consistent with the notion of 𝜎

𝑜as

a friction stress belowwhich dislocations will notmove in thematerial in the absence of grain boundaries. For𝐷

𝑀= 2, 𝑑 =

0.03 𝜇m (see (5)) and 𝜎𝑦= 4,387.84MPa. From a theoretical

point of view (see (6)), this microstructure can be related tosuch a high value of 𝜎

𝑦. Truly, a loss of strengthening for


2.0

500

400

300

200

100

0

Yield

stre

ss𝜎y

(MPa)

1.83 3.0 4.0 5.0 6.0 7.0

1.0 1.10 1.20 1.30

DF

DF

∞

(1.30 − DF)−1/2

(1.30 − DF)−1/2

Figure 7: Hall-Petch type relations between yield stress and fracturesurface fractal dimension for AISI 316L austenitic stainless steel.

𝑑 = 0.03 𝜇m (30 nm) which falls into the so-called “inverseHall-Petch dependence zone” [51, 52] should occur. For thecurve𝜎

𝑦versus𝐷

𝑀in Figure 8, two arithmetic subscales have

been introduced which encompass the full theoretical rangeof 𝐷𝑀

(1 ≤ 𝐷𝑀≤ 2). The natural link between the fractal

dimension of grain boundaries and mechanical propertieshas been proven to be very important in metallic materialsengineering [53–55]. On the other hand, the Hall-Petch typerelationship between 𝜎

𝑦and 𝐷

𝑀(see (6)) facilitates the

comprehension of this link.

3.3. Relation between 𝐷𝐹and 𝐷

𝑀. The relation between the

fracture surface fractal dimension 𝐷𝐹and the microstruc-

tural fractal dimension𝐷𝑀can be found by equating (2) and

(5), which leads to

𝐷𝐹= 𝜆−𝛽 (𝛼)

1/𝜂(𝐷𝑀)−1/𝜂 (7)

which in turn, taking into account the values of the constants𝛼, 𝛽, 𝜂 and 𝜆, gives

𝐷𝐹= 1.30− 162.74 (𝐷

𝑀)−22.22, (8)

where 22.22 ≈ 1/0.045 and 162.74 is a constant withoutdimensions. The relation between𝐷

𝐹and𝐷

𝑀is represented

in Figure 9 and compared with𝐷𝐹= 𝐷𝑀. As can be seen, the

values of𝐷𝐹are smaller than the values of𝐷

𝑀, being the limit

values: 𝐷𝑀= 1.328 for 𝐷

𝐹= 1 and 𝐷

𝑀= 2 for 𝐷

𝐹= 1.30.

The experimental values for 𝐷𝑀and 𝐷

𝐹ranged between the

intervals 1.365 ≤ 𝐷𝑀≤ 1.504 and 1.142 ≤ 𝐷

𝐹≤ 1.281 as

have been quoted in Tables 1 and 2, respectively.The very nature of the relationship between 𝐷

𝑀and

𝐷𝐹possesses great difficulties in analysis. Nevertheless, the

present results confirm that an increase in𝐷𝑀or𝐷𝐹involves

an increase in the yield stress as the grain size becomessmall. Although both the box counting method [42], which

Yield

stre

ss𝜎

4,500

4,000

300

200

100

01.0 10 30 50 70 90 2,000 2,212

(DM)1/(2𝜂)

DM

DM

(DM)1/(2𝜂), (𝜂 = 0.045)

1.0 1.2 1.4 1.98 1.99 2.00

y (M

Pa)

Figure 8: Hall-Petch type relations between yield stress andmicrostructural fractal dimension for AISI 316L austenitic stainlesssteel.

2.0

1.9

1.3

1.2

1.1

1.0

Frac

ture

surfa

ce fr

acta

l dim

ensio

nD

F

DF = DMDF = 1.30 − 162.74 (DM)

−1/𝜂, (𝜂 = 0.045)

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Microstructural fractal dimension DM

1.3

Figure 9: Microstructural fractal dimension 𝐷𝑀

versus fracturesurface fractal dimension𝐷

𝐹as compared with𝐷

𝐹= 𝐷𝑀.

has been used by Colás [36] to determine 𝐷𝑀, and the slit-

island method [17, 19, 24, 38, 39, 41], used in the presentwork to determine𝐷

𝐹, are based on 2Dmetallographic image

obtained by grinding the specimen surface flat, the kind ofmicrostructures in which they were applied is essentiallydifferent. Note that the different methods to determine 𝐷

𝐹

are, in theory, equivalent, but the SIM method was selectedbecause it is more suitable for the analysis of a rough surface.In addition, a great part of the 𝐷

𝐹data in the literature is

based on this method, which facilitates comparison. In spiteof the above, the changes in𝐷

𝐹and𝐷

𝑀for the corresponding

range of grain sizes were the same:Δ𝐷𝐹= (1.281)−(1.142) =

0.139 and Δ𝐷𝑀= (1.504) − (1.365) = 0.139 (Tables 1

and 2). Although these results can be regarded as fortuitous,they suggest, in principle, that the increase in complexity


experienced by the net of grains between 1194∘C and 904∘C(increasing the value of 𝐷

𝑀) was completely translated into

the respective fracture surface. No previous results for thecorrelation between 𝐷

𝑀and 𝐷

𝐹exist for AISI 316L stainless

steel, which makes a comparison difficult to achieve.Estimating the influence of the microstructural fractal

dimension on the fracture surface fractal dimension can bevery important, theoretically and in practical applications.The connection between fractal characteristics of materialsand mechanical properties can be easily established throughequations such as (4) and (6).

4. Conclusions

From the present study, it can be seen that the fracturesurface fractal dimension 𝐷

𝐹and the fractal dimension of

the grain microstructure𝐷𝑀in AISI 316L austenitic stainless

steel can be related to the strength of the material throughthe Hall-Petch law, which provides a well-established andsound platform to study and analyze this relation.The presentresults indicate the strong interplay between micrograins(microstructure), yield stress (mechanical property), andfracture topography (fracture behavior) for the studiedmate-rial. The increase in complexity of the microstructure asthe grain size decreases is measured by 𝐷

𝑀and translated

into the fracture surfaces, which become more irregular asindicated by the values of 𝐷

𝐹. The relation between grain

size,𝐷𝑀, mechanical properties, and𝐷

𝐹, developed for AISI

316L steel, should be generalized and applied to most of thecommercial metallic alloys of technological importance.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to thank the staff of the ElectronMicroscopy Center of the Faculty of Science of the CentralUniversity of Venezuela for their assistance, the Ferrum C.A.for providing test material, and the Scientific andHumanisticDevelopment Council of the Central University of Venezuelafor financial support.

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Research Article Microstructure, Strength, and Fracture … · 2017. 7. 24. · InternationalJournalofMetals 1.35 1.30 1.25 1.20 1.15 1.10 r2 = 0.99 Average grain size Fracture surface